322 XV NORMED ALGEBRAS AND SPECTRAL THEORY are analytic in the interior of D (15.1.5). We shall show that the characters of A are again in this case the restrictions to A of the Dirac measures sx(x e D). From this it will follow (15.3.7) that xh^ex is again a homeomorphism of D onto the spectrum X(A). Let/0 be the canonical injection of D in C ; then it is enough to show that/0 and the unit element 1 of A generate a dense subalgebra of A (for it is clear that SpA(/0) = D, and the result will then follow by (1 5.3.6)). Now, let A0 denote the subalgebra of A consisting of the functions which are restrictions to D of functions analytic on some neighborhood of D in C. This subalgebra is dense in A, because for each integer n > 0 and each function /e A, the function fn which is the restriction to D of the mapping of ((n + l)/w)D into C clearly belongs to A0, and/rt -»/ uniformly on D as n~+ +00 (3.16.5). On the other hand, every function /eA0 is the uniform limit (in D) of a sequence of polynomials Pn , namely the partial sums of the Taylor series for / at the point 0 (9.9.1 and 9.9.2). Hence the set of poly- nomials in C is dense in A, and our assertion is proved. We remark that here the Gelfand transformation is isometric, but ^(A) is distinct from (15.3.9) Let B be the Banach algebra consisting of the restrictions to the unit circle U : |C| = 1 of the functions belonging to the algebra A of (15.3.8). Then the restriction mapping/W/| U is an isometric isomorphism of A onto B : this is an immediate consequence of the maximum modulus principle (9.5.9), which implies that ||/|| = ||/| U||. Here the mapping xi-+ex of U into X(B) is again continuous and injective; but it is no longer a homeomorphism, because all the characters of A can be identified with characters of B, and therefore X(B) = X(A) = D. PROBLEMS 1. Let X be a metrizable compact space and B a subalgebra of ^C(X) containing the unit element. Assume that B is equipped with a norm ||jt ||B which makes it a Banach algebra. (a) Show that ||JC||B ^ W (the norm on ^C<X))» bV remarking that <p(t) :x^x(t) is a character of B for all / e X. Deduce that 0 is the only quasi-nilpotent element of B. (b) Show that 9? is a continuous mapping of X into X(B). If B separates the points of X (7.3) then 99 is a homeomorphism of X onto a closed subset of X(B). (c) Suppose that, for each function x e B, the conjugate x belongs to B, and that if x E B is such that x(t) ^ 0 for all t e X, then the inverse x"1 of x in ^C(X) belongs to B. Show that in these conditions the mapping 99 is surjective. (Let m be a maximal ideal of B, and let Z be the set of all t e X such that x(t) — 0 for all x E m. Show that in C, and let A = ^(D) be the